Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $x = \dfrac{p^2 - 6p - 40}{-2p^2 - 8p + 90} \div \dfrac{-5p - 20}{p + 9} $
Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{p^2 - 6p - 40}{-2p^2 - 8p + 90} \times \dfrac{p + 9}{-5p - 20} $ First factor out any common factors. $x = \dfrac{p^2 - 6p - 40}{-2(p^2 + 4p - 45)} \times \dfrac{p + 9}{-5(p + 4)} $ Then factor the quadratic expressions. $x = \dfrac {(p + 4)(p - 10)} {-2(p + 9)(p - 5)} \times \dfrac {p + 9} {-5(p + 4)} $ Then multiply the two numerators and multiply the two denominators. $x = \dfrac { (p + 4)(p - 10) \times (p + 9)} { -2(p + 9)(p - 5) \times -5(p + 4)} $ $x = \dfrac {(p + 4)(p - 10)(p + 9)} {10(p + 9)(p - 5)(p + 4)} $ Notice that $(p + 9)$ and $(p + 4)$ appear in both the numerator and denominator so we can cancel them. $x = \dfrac {(p + 4)(p - 10)\cancel{(p + 9)}} {10\cancel{(p + 9)}(p - 5)(p + 4)} $ We are dividing by $p + 9$ , so $p + 9 \neq 0$ Therefore, $p \neq -9$ $x = \dfrac {\cancel{(p + 4)}(p - 10)\cancel{(p + 9)}} {10\cancel{(p + 9)}(p - 5)\cancel{(p + 4)}} $ We are dividing by $p + 4$ , so $p + 4 \neq 0$ Therefore, $p \neq -4$ $x = \dfrac {p - 10} {10(p - 5)} $ $ x = \dfrac{p - 10}{10(p - 5)}; p \neq -9; p \neq -4 $